OF BLACKFOLD KENTARO TANABE (UNIVERSITY OF BARCELONA) - PowerPoint PPT Presentation

SELF-GRAVITY EFFECTS OF BLACKFOLD KENTARO TANABE (UNIVERSITY OF BARCELONA) collaboration with R. Emparan and S. Kinoshita

1. INTRODUCTION blackfold approach effective worldvolume theory for dynamics of black brane 𝑆 ≫ 𝑠 0 curvature effects ~ 𝑠 0 𝑆 𝑆 Application • construction of higher dimensional black holes [Emparan et.al. (2007,2009)] • stability analysis of black brane [Camps, et.al. (2009)] • D-brane probe in thermal background [Grignani, et.al. (2011), Arms, et.al. (2012),..]

BLACKFOLD AS BLACK HOLE black hole solution can be constructed by gluing branes Matched Asymptotic Expansion various properties of higher dimensional black holes 1. possible topology of black holes phase diagram of solutions,… 2.

PRESENT STATUS possible topology phase diagram Note: these studies were done by test brane analysis

PURPOSE Our purpose: • clarify its matching structure how solution is constructed by blackfold approach • computing back reaction ( self-gravity effects ) check the validity of blackfold approach construction of more precise phase diagram

2. MATCHING LADDER

BLACK RING AS BLACKFOLD Black ring is constructed from black string ( black 1-brane ) schematic black ring boosted black string bending 𝑠 0 𝑆 matched asymptotic expansion match flat spacetime black string + perturbation + perturbation near region far region

BLACKFOLD EQUATION brane should satisfy regularity condition ( “no tension” ) 𝜍 = 0 𝑈 𝜈𝜉 𝐿 𝜈𝜉 brane’s energy momentum tensor extrinsic curvature This condition guarantees the regularity of the black hole horizon and determines possible topology [Camps and Emparan (2011)]

FIRST MATCHING First order solution at far region ( Newton solution ) 𝑆 boosted black string constitute the 𝜀 – function source at far region 𝜍 = 0 𝑈 𝜈𝜉 𝐿 𝜈𝜉

MATCHIG AT NEAR REGION bending the black string excites the perturbation on the black string 𝑧 𝜍 black string perturbation = 𝜍 𝑧 𝜍 𝑕 𝜈𝜉 + 𝐿 𝜈𝜉 𝑚 : spherical harmonics From dimensional analysis, 𝐵 𝑚 𝑠 𝑚 0 𝑆 𝑚 mode perturbation = Amplitude is determined by far region solution

MATCHING AT FAR REGION perturbation on black string also excite new energy momentum tensor at far region 𝐵 𝑚 𝑠 𝑚 𝑒−4 1 − 𝑠 0 0 𝑆 𝑠 𝑒−4 excited by far region solution create new energy momentum tensor at far region

MATCHING LADDER far region = flat + perturbation 𝑠 𝑚 0 𝑆 Newton solution correction to energy momentum tensor 𝜍 = 0 𝑈 𝜈𝜉 𝐿 𝜈𝜉 𝑚 mode perturbation black string 𝑚 𝑠 0 𝑆 ( at least ) near region = black string + perturbation

3.SELF-GRAVITY EFFECTS

SELF-GRAVITY using the matching ladder, compute the self-gravity effects self-gravity order ∆ Ψ (2) = Ψ 2 A B Newton solution self-gravity correction Ψ 2 = 𝑃 𝑠 𝑒−4 0 𝑆

SECOND ORDER For example, consider 6-dimensions case 𝑠 2 0 𝑆 Newton solution self-gravity correction 𝜍 = 0 𝑈 𝜈𝜉 𝐿 𝜈𝜉 𝑚 = 1 mode 𝑚 = 0 mode black string 𝑠 1 𝑠 2 0 0 𝑆 𝑆 corrections to mass, angular momentum and area

CORRECTIONS Solving and matching the perturbations, the self-gravity effects shows 𝜌 2 𝑁 = 16𝐻 𝑆𝑠 mass 0 𝐾 = 𝜌 self-gravity correction 4𝐻 𝑆 2 𝑠 2 angular momentum 0 3 1 + 𝐷 1 + 1 4 log 𝑠 𝑠 2 0 0 𝑇 = 𝜌 6 𝑆𝑠 area ( 𝐷 1 < 0 ) 0 𝑆 𝑆 up to trivial changing ( 1 st law of black hole )

PHASE DIAGRAM Phase diagram is useful to see the physical effects of self-gravity 1 st order solution 𝑡 𝑒−3 = 𝑡 0 𝑇 𝑒−3 𝑁 𝑒−2 𝑘 𝑒−3 = 𝑘 0 𝐾 𝑒−3 𝑁 𝑒−2 2 nd order solution (self-gravity )

4. SUMMARY • complete the matching ladder of blackfold approach (black string case) • compute the self-gravity effects future work • matching ladder of general blackfold • incorporating the intrinsic perturbation